数理解析研究所講究録
第 1737 巻 2011 年 91-104

91

DEVELOPMENTS IN NONCOMMUTATIVE KOROVKIN-TYPE
THEOREMS

M. N. N. NAMBOODIRI
Department of Mathematics,
Cochin University of Science and Technology, Kochi-682022.
Email: nambu@cusat.ac.in, nbdri@yahoo.com

1. INTRODUCTION
The classical approximation theorem due to P. P. Korovkin [11] in 1953,
unified many existing approximation processes such as Bemstein polynomial approximation of continuous functions. Korovkin’s discovery inspired
many researchers that lead to Korovkin-type theorems and Korovkin sets
in various settings such as more general function spaces, Banach algebras,
Banach lattices and operator algebras. Another major advancement was thediscovery of geometric theory of Korovkin sets by Y. A. \v{S}a\v{s}kin [21] and D.
E. Wulbert in 1968 [7]. A detailed survey of most of these developments
can be found in the article of Berens and Lorentz in 1975 [7], monographof Altomare and Campiti [1] most recent survey by Aitomare [2] whichcontains several new results also.
This article aims at providing a rather short survey of the developmentsin the so called noncommutative Korovkin-type approximation theoly andKorovkin sets (quantization of Korovkin theorems, W. B. Arveson [6]) in
the settings ofand-algebras. Due to technical reasons only important theorems are quoted that too without proofs. However an attempt ismade to provide illustrative examples, a few new results and research problems. First we quote three major theorems due to Korovkin following thearticle of Berens and Lorentz [7]. We designate these as Korovkin’s typeI, type II and type III theorems. The survey will be about ‘quantization’ ofthese three theorems!
$C^{*}$$W^{*}$

Type I Korovkin’s theorem. Let $\{\Phi_{n} : n=1,2,3, \ldots\}$ be a sequence of
positive linear maps on $C[a, b]$ and for each of the functions $g_{k}(x)=x^{k}$ ,$x\in[a, b],$ $k=0,1,2$ , let$\lim_{narrow\infty}\Phi_{n}(g_{k})=g_{k}$uniformly on $[a, b],$ $k=0,1,2$ .

Then$\lim_{narrow\infty}\Phi_{n}(f)=f$

uniformly on $[a, b]$ for all$f$

in $C[a, b]$ .

Definition 1.1. A set in $C[a, b]$ is called a test set or Korovkin set for
positive linear operators on $C[a, b]$ if for every sequenceof positive$S$$\{\Phi_{n}\}$92
M. N. N. Namboodiri

linear operators on $C[a, b],$in implies that
$S$

$s$

$\lim_{narrow\infty}\Phi_{n}(f_{=}f$Type I theorem says thatuniformly on $[a, b]$ for every
uniformly of $[a, b]$ for all $f\in C[a, b]$ .

$\lim_{narrow\infty}\Phi_{n}(s)=s$$\{1, x, x^{2}\}$is a test set.

Type II Korovkin’s theorem. There is not test set for $C[a, b]$ consisting
only of two functions. Thus the cardinality of a test set is atleast 3.
Type In Korovkin’s theorem. A triple
is a test set of $C[a, b]$].
exactly when it is asystem on
This article is divided into four sections. The next three sections are
devoted to noncommutative Type I, Type II and Type III theorems. The
last section contains some aspects of weak Korovkin type theorems and its
geometric formulation.
$\{f_{0}, f_{1}, f_{2}\}$$\check{C}eby\check{s}ev$

$a,$

$b$

2. TYPE I THEOREMSThis section deals with type I Korovkin theorems in the settings ofalgebras. It is assumed that the -algebras considered here are over complex numbers and always contain identity unless otherwise specified.
$C^{*}-$$C^{*}$

be complex
Deflnition 2.1. Let $d$ and-algebras with identities
andbe a positive linear contraction (arespectively and let :. For alinear map that preserves positivity and such that$H$
,Of
$K_{+}(H,T)$
subset of the Korovkin closure
is defined as$C^{*}$$\mathscr{B}$$T$$1_{\mathscr{D}}$$1_{d}$$darrow \mathscr{B}$$T$$T(1_{d})\leq 1_{\mathscr{D}})$$\{a\in d|\lim_{\alpha}\Phi_{\alpha}(a)=T(a)$for every net$\{\Phi_{\alpha}\}_{\alpha\in I}$of positive linear contractions fromto$\mathscr{B}$such that$\lim_{\alpha}\Phi_{\alpha}(h)=T(h)$for all$h$in $H$ }Here convergence considered is the norm convergence unless otherwise
stated explicitly. It seems that the first noncommutative Korovkin type theorem was due to W. B. Arveson in 1970 for -homomorphisms where .Of
is $C(X)$ , the-algebra of all complex continuous functions on a compact, Hausdorff space $X$ . We recall this, being the first of its kind. Recall
that for a subset $H$ of $C(X)$ , the Choquet boundary
is defined as$X$
the set of all points inhassuch that the evaluation functionals
the unique positive linear extension
to $C(X)$ . Also the supportof$X$:
is defined as set of all in such that $f(x)=0$ whenever$*$$C^{*}$$\partial_{H}^{+}(X)$$\epsilon_{x}|H$$x$$\epsilon_{X}$$T$$C(X)arrow \mathscr{B}$$K_{T}$$x$$T(f)=0$ .

2.1. Theorem. Let $H$ be a subset of $C(X)$ containing 1 and let $T:C(X)arrow$be a homomorphism such that
. Then $K_{+}(H, T)=$
$C(X)$ .
The proof of the above theorem uses the lattice theoretic properties of theselfadjoint part of $C(X)$ . It is to be recalled that the selfadjoint pall of a-algebra
is a lattice in the natural order if and only of .Of is commutative. In 2009, W. B. Arveson published a paper [6] in which he worked$\mathscr{B}$$C^{*}$$*$$K_{T}\subseteq\partial_{H}^{+}(X)$93
out a relation between ‘noncommutative Choquet boundary’ and ‘hyperrigid subspaces’ (Korovkin sets) for general
algebras. In what follows abrief sketch of the developments from 1970 to 2010 is provided. Important
theorems that appeared in the articles published during the above period byvarious authors, are regarding the following two questions.
(1) When does the Korovkin closure has an algebraic stmcture(2) When is the Korovkin closure the full
algebra. Mainly four typesof maps are considered in this settings. A linear map :is called(a) Positive if
is positive for all $x\in A$(b) Schwarz map if $\Phi(x^{*}x)\geq\phi(x)^{*}\Phi(X)$ for all $x\in$(c) Completely positive if:
is a positive
for all positive integers , whereis the set of all
matricesover andis the map on
defined by$C^{*}$$C^{*}$$\Phi$$arrow \mathscr{B}$$\Phi(x^{*}x)$$\Phi^{(n)}$$\otimes M_{n}(\mathbb{C})arrow \mathscr{B}\otimes M_{n}(\mathbb{C})$$n$$n\cross n$$M_{n}(\mathbb{C})$$\Phi^{(n)}$$\mathbb{C}$$\otimes \mathscr{B}$$\Phi^{(n)}(a_{ij})=(\Phi(a_{ij}))$, where $(a_{ij})\in\otimes M_{n}(\mathbb{C})$is contractive for each positive integer .
The main tools being used in the development of the commutative theory
for positive linear maps are Kadison-Schwarz type inequalities and Choquet
boundary theory. It is to be mentioned that every completely positive map ofnorm
is a Schwarz map and the very definition of Schwarz map implies
the Schwarz type inequality. For general positive linear map :with norm
Kadison proved that(d) Completely contractive if$\Phi^{(n)}$$n$$\leq 1$$\Phi$$arrow \mathscr{B}$$\leq 1$$\Phi(x^{2})\geq\Phi(x)^{2}\forall x\in,$ $x^{*}=x$.

This fundamental inequality is known as Kadison-Schwarz inequality andhas been improved by many mathematicians like M. D. Choi [9] and T.
Fumta [10]. These improvements will have some effect on the study of
Korovkin sets. However this possibility is yet to be investigated.
Using Kadison-Schwarz inequality, W. M. Priestley [17] proved the following theorem in 1976.

2.2. Theorem. Let be a
linear maps on such that

$C^{*}$-algebra and let$\Phi_{\alpha}(1_{\ovalbox{\tt\small REJECT}})\leq 1_{\ovalbox{\tt\small REJECT}}$$\{\Phi_{\alpha}\}_{\alpha\in I}$$\forall\alpha\in I$

be a net of positive.

Then the set
$J$

is a

$:= \{x\in|\lim_{\alpha}\Phi_{\alpha}(a)=a\forall a\in\{x, x^{*}\circ x, x^{2}\}\}$-algebra in .
Recall that a -algebra in
is a norm closed, closed subset ofwhich is also closed under the Jordan product , namely$J^{*}$$J^{*}$$*$$0$$a\circ b=ab+ba,$$a,$$b\in$For Schwarz maps A. G. Robertson [18] proved the following theorem.

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M. N. N. Namboodiri

2.3. Theorem. Let be a
be a net of Schwarz maps on$C^{*}$-algebra with identity
such that.
,$1_{d}$and let$\{\Phi_{\alpha}\}_{\alpha\in I}$$\forall\alpha\in I$$\Phi_{\alpha}(1_{d})\leq 1_{d}$Then the subset$K= \{x\in|\lim_{\alpha}\Phi_{\alpha}(a)=a$for $a\in\{x,$$x^{*}x,$ $xx^{*}\}\}$

is a -algebra in $d$ .
Subsequently B. V. Limaye and M. N. N. Namboodiri [12] improved theresults of Priestley and Robertson to obtain the following theorem in 1982:$C^{*}$

2.4. Theorem. Let
respectively. Let
such thatandbe -algebras with identities
be a net of positive linear maps from $d$ toand$\{\Phi_{\alpha}\}_{\alpha\in I}$$\mathscr{B}$$C^{*}$$\Phi_{\alpha}(1_{d})\leq 1_{\ovalbox{\tt\small REJECT}}$

Let

:

$T$

$arrow \mathscr{B}$

$J$

is a

$1_{d}$$\forall\alpha\in I$$1_{\mathscr{D}}$$\mathscr{B}$.

be $a*$ -homomorphism. Then the subset$:= \{x\in|\lim_{\alpha}\Phi_{\alpha}(a)=T(a)$, for $a\in\{x,$ $x^{*}ox\}\}$-algebra in .Of.
If allare Schwarz maps and $Ta*$ -homomorphism, then isa -algebra in .
In the above cases the test set was symmetric with respect to operation.
For the case when this symmetry is not assumed, B. V. Limaye, M. N.
N. Namboodiri in 1984 [14] and A. G. Robertson in 1986 [19] proved thefollowing theorems.
It is known that Korovkin type approximation theory leads to deeper understanding of the stmcture under consideration. The following interesting
theorem in [14] reveals this.
$J^{*}$$\Phi_{\alpha},$$J$$\alpha\in I$$C^{*}$$*$

2.5. Theorem. Let$1_{d}$$=\mathscr{B}$be a noncommutative$C^{*}$

algebra with identity

and let$D= \{x\in d|\lim_{\alpha}\Phi_{\alpha}(a)=a$

for all $a\in\{x,$$x^{*}x\}\}$. Then $D$ iswhereis a net of Schwarz maps on $d$ with norm$D$, the seta subalgebra of $d$ . Also is closed if and only if .Of
of allmatrices over .
behaves like a commutative
The above theorem shows thatalgebra and this is the only noncommutative one! In fact, Limaye andNamboodiri proved that, among all finite dimensional noncommutativeis the only one for which $D$ in Theorem 2.5 is closed.
algebraRobertson proved that finite dimensionality assumption can be dropped.
We bypass several important developments of Korovkin approximation
theory for commutative as well as noncommutative Banach algebras. Excellent exposition can be found by Micheal Panneberg, [1, Appendix $A$ ],Ferdinand Beckhoff [1, Appendix ] and F. Altomare [2]. However M.
Uchiyama’s paper [23] give estimates the norm related to Schwarz map. He$\leq 1$$\{\Phi_{\alpha}\}_{\alpha\in I}$$=M_{2}(\mathbb{C})$$*$$2\cross 2$$\mathbb{C}$$C^{*}-$$M_{2}(\mathbb{C})$$C^{*}$$*$$M_{2}(\mathbb{C})$

$B$

95
al so obtains several extensions of Korovkin type theorems by using operator
monotone functions and T. Ando’s inequality. In this paper he also unified
several earlier results by introducing -subalgebras and the associated generalized Schwarz maps with respect to the product . More over the proofs
that he gives are simpler than the earlier ones.
For the sake of completion we quote a couple of theorems of Uchiyama[23] for the
algebra $C(X)$ .
$0^{*}$$0$$C^{*}$

2.6. Theorem. [23, Theorem 3.1]

Let $S\subset C(X)$ and $C^{*}(S)$ be the -algebra generated by . Let bea operator monotone function defined onsuch that $f(O)\leq 0$ and$f(\infty)=\infty$ . Set $g=f^{-1}$ then we have
$S$

$C^{*}$$f$$[0, \infty]$$C^{*}(S)\subseteq K_{C(X)}(SU\{g(|u|^{2}) :u\in S\})$

if $f(0)=0$ or $1\in S$ . Where the set on the right side denotes the Korovkinclosure.

2.7. Theorem. [23, Theorem 2.12] Let
be a sequence of Schwarz
maps from to where and are -algebras with identities, andlet :be $a*$ -homomorphism. Let be an operator monotonefunction onwith $f(0)=0,$ $f(\infty)=\infty$ . Set $g=f^{-1}$ . Then the set$\{\Phi_{n}\}$$\mathscr{B}$$C^{*}$$\mathscr{B}$$arrow \mathscr{B}$$\Phi$$f$$[0, \infty]$$C=\{a\in|\Phi_{n}(x)arrow\Phi(x)$for $x=a,$ $g(a^{*}a)$ and $g(aa^{*})\}$

is a -subalgebra.
Another important development was the use of Krein-Millman theorem
for compact convex sets and the associated unique extension property. For
noncommutative -algebras, the following theorem was proved by Takahasi in 1979 [22].
$C^{*}$$C^{*}$

2.8. Theorem. Let be an extreme state of a algebra and let $x\in_{+}$
peaks for , that is the supports of and in the enveloping von Neumann
algebra add up to 1. Let
be a net of positive linear functionals of$\lim_{\alpha}\phi_{\alpha}(1-)=1$
and assume thatand
. Then we$C^{*}$$\phi$$x$$\phi$$\phi$$\{\phi_{\alpha}\}_{\alpha\in I}$$\lim_{\alpha}\phi_{\alpha}(x)=0$

have$\lim_{\alpha}\phi_{\alpha}(a)=0$

for all$a$om .Of.

Though the above theorem is elegant, for an arbitrary -algebras there is
no way of finding extreme states (or pure states), where as for commutative
algebras extreme states are point evaluations. In the case of -algebra$B(H),$ $H$ a Hilbert space,a vector state better results are known to exist.
See for example Limaye-Namboodiri (1979) [1, Appendix B.], Altomare
1987 [1] and Dieckmann 1992 [1].
$C^{*}$$C^{*}$$C^{*}$$\phi$

2.9. Theorem (Altomare [1]). Let $T\in B(H)$ , be a non zero compact
operator, a simple eigenvalue of such thata corresponding
unit eigenvector and let
be the vector state corresponding to .
Put
. Then$T$$\lambda$$\phi=\langle\cdot x,$$\Vert T\Vert=|\lambda|,$ $x$$x\rangle$$S=I_{H}+ \frac{1}{|\lambda|^{2}}T^{*}T-\frac{1}{\lambda}T_{\lambda}^{1}-arrow T^{*}$$K_{+}(\{I_{H}, S\}, \phi)=B(H)$ .

$x$

96
M. N. N. Namboodiri

Heredenotes the identity operator on $H$ .
The following theorem is due to Dieckmann$I_{H}$

2.10. Theorem (Dieckmann). Let $T\in B(H)$ be a strictly positive compact operator on $H$ and let $d(H)=C^{*}(I_{H}, K(H))$ , where $K(H)$ is theset of all compact operators on $H$ . Let be the complex homomorphism on$K\in K(H)$ . Then $K_{+}(\{I_{H}, T\}, \phi)=$$(H)$ defined by$\phi$$\phi(\lambda I_{H}+K)=\lambda,$$(H)$ .
Finally we go through Arveson’s contributions to noncommutative Korovkin type theorems and Korovkin sets in 2009 [6] via his own theory of
noncommutative Choquet boundary theory of operator systems. In the fundamental papers during 1969-70 and 2008, he introduced and proved many
concepts and the theorems related to non commutative Choquet boundary
and Silov boundary ideals $co$lTesponding to operator systems. This is quite
analogous to classical theory of Choquet and Silov boundaries for function systems. Analogous to the work of Saskin, Arveson studied the relation between noncommutative Korovkin sets and noncommutative Choquet
boundary in 2009 [6]. He proved many interesting theorems in this settings,
though some of these results were already known to exist. We start with thealgebras [6].
notion of hyperrigid set of generators of$C^{*}$

2.11. Definition. A finite of countably infinite set of generators of algebra .Of is said to be hyperrigid if for every faithful representation$B(H)$ of al on a Hilbert space $H$ andsequence of unit preserving$B(H)arrow
B(H),$ $n=1,2,3,$:
completely positive maps (UCP)$G$$C^{*}$$\pi()\subseteq$$evel\gamma$$\Phi_{n}$$\ldots$$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(g)-\pi(g))\Vert=0\forall g\in G\Rightarrow\lim_{narrow\infty}\Vert\Vert\Phi_{n}(\pi(a)-\pi(a))\Vert=0$$\forall a\in$,. He then proves the following basic theorem.

2.12. Theorem. For
separable operator system that generates a,
the following are equivalent.
-algebra(i) is hypemigid(ii) Fornon degenerate representation : $arrow B(H)$ on a separable: $arrow B(H)$
Hilbert space $H$ and every sequenceof UCP maps;$S$$evel\gamma$$C^{*}$

$S$

$\pi$

$evel\gamma$$\Phi_{n}$$\lim_{narrow\infty}\Vert\Phi_{n}(s)-\pi(s)\Vert=0$$\forall s\in S\Rightarrow\lim_{narrow\infty}\Vert\Phi_{n}(a)-\pi(a)\Vert=0$

for all $a\in d$ .
(iii) For every non degenerate representation : $arrow B(H)$ on a separable
Hilbert space, $\pi/S$ has the unique extension property. That is, $\pi/S$ has aunique completely positive linear extension to ,Of.
(iv) For every unital
algebra , every unital homomorphism ofalgebras :and every UCP map ‘ :$\pi$$C^{*}$$\theta$$arrow \mathscr{B}$$\Phi(x)=x$$C^{*}-$$\mathscr{B}$$\Phi$$\mathscr{B}arrow \mathscr{B}$$\forall x\in\theta(S)\Rightarrow\Phi(x)=x\forall x\in\theta(d)$.

One of the main results (Theorem 3.3 [6]) that Arveson obtains as a consequence of the above theorem, is known to exist. In fact much better theorem

97
can be found in [6]. However he proves a5.1] which is as follows.
2.13. Theorem. Let$C^{*}$$S$

strong theorem [6, Theorem$vel\gamma$

be a separable operator system whose generated-algebra ,Of has countable spectrum such that $evei\gamma$ irreducible represen-tation of

is a boundary representation for . Then
$S$

$S$

is hyperrigid.

2.14. Arveson’s conjecture. If every irreducible representation of is aboundary representation for a separable operator system , then is hyper$S$$S$

rigid.
Now recall that the following theorem was proved by Y. A. Saskin for
positive linear contractions and D. E. Wulbert for linear contractions [7].

2.15. Theorem. Let

be a subset of $C(X)$ that separates points of $X$
and contains the constant function. Then is a Korovkin set for linear$G_{0}=$
contractions or positive linear contractions if and only if
$G$

$G$

$1_{X}$$\partial_{Ch}G_{0}=X,$

span .
The noncommutative Choquet boundary was defined by Arveson [6] inthe following way.
$G$

2.16. Definition. Let

be an operator system in a-algebra , i.e., aself adjoint linear subspace of$1_{d}\inS$such thatand $=C^{*}(S)$the-algebra generated by and. A boundary representation foris an $i_{lT}educible$ representation ofsuch that $\pi/S$ has a unique com$d$
pletely positive linear extension to . The set of all unitary equivalenceclasses of all boundary representations for id defined as the noncommutative Choquet boundary of the operator system .
Since $i_{lT}educible$ representation of the function space $C(X)$ can be identified with points in $X$ itself, Arveson’s notion of Choquet boundary for
operator systems is an exact noncommutative analogue of the classical one
for function systems.
In what follows we examine the possibility of extending Arveson’s theorem quoted here for linear contractions. Since extension theorem for completely positive maps is not available, we need to define hypernigidity separately. So we introduce strong hyperrigidity so as to suit completely con$C^{*}$$S$$C^{*}$$S$$S$$1_{d}$$\pi$$\partial_{S}$

$S$

$S$

tractive maps.

2.17. Definition. A finite or countably infinite setof generators of aalgebrais said to be strongly hyperrigid if for every faithful representation of in $B(H)$ and for every sequence
completely contractive
maps from $\pi()$ to $B(H)$
$\pi$

$G$

$C^{*}-$$\Phi_{n}$$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(g))-\pi(g)\Vert=0$$\forall g\in G$$\Rightarrow\lim_{narrow\infty}\Vert\Phi_{n}(\pi(a))-\pi(a)\Vert=0$$\forall a\in$.

98
M. N. N. $Nambood\ddot{m}$2.18. Remarks. It can be seen that the strong hyperrigidity coincide withhyperrigidity for UCP as a consequence of Arveson’s extension theorem for
CP maps. However hyperrigidity of need not imply strong hyperrigidity.
To overcome this difficulty it would be reasonable to assume that is closed
under operation if necessary.
In what follows we aim at identifying ‘obstmctions’ to strong hyperrigidity. We also assume that the operator system is closed and contains identity element.
$G$

$G$

$*$

$*$

2.19. Characterisation theorem. For separable operator system thatgenerates a -algebra ,Of, the following are equivalent:(i) is strongly hyperrigid.
: $arrow B(H)$ on a sep(ii) For every non degenerate representation: $darrow B(H)$ completely
arable Hilbert space and every sequencecontractive maps,$S$$C^{*}$

$S$

$\pi$

$H$

$\Phi_{n}$$\lim_{narrow\infty}\Vert\Phi_{n}(s)-\pi(s)\Vert=0$$\forall s\in S$$\forall a\in$$\Rightarrow\lim_{narrow\infty}\Vert\Phi_{n}(a)-\pi(a)\Vert=0$.

: $darrow B(H)$ on a sepahas the unique extension property.
rable Hilbert space(iv) For
unital -algebra , every unital homomorphism of) mapand every (for every UCP map :algebras :(iii) For every non degenerate representation$\pi$$H,$ $\pi/S$$C^{*}$$evei\gamma$$\theta$$C^{*}-$$\mathscr{B}$$darrow \mathscr{B}$$\Phi$$\theta(d)arrow \mathscr{B}$$\Phi\thetaarrow \mathscr{B}$$\Phi(x)=x$$\forall x\in\theta(S)\Rightarrow\Phi(x)=x$$\forall x\in\theta(d)$.

Proof. The proof is more or less the same as that of Arveson. However thedetails are provided. We show that$(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(iv)\Rightarrow(i)$.

$(i)\Rightarrow(ii)$on a: $arrow B(H)$ be a non degenerate representation ofLet: $darrow B(H)$ be a sequence ofseparable Hilbert space $H$ and let(completely contractive) linear maps such that$\pi$$\Phi_{n}$$\lim_{narrow\infty}\Vert\Phi_{n}(s)-\pi(s)\Vert=0$$\forall s\in S$.

Let : $arrow B(H)$ be a faithful representation of on another separable
Hilbert space $K$ .
: $arrow B(K\oplus H)$ is a faithful representation of $d$ on
Then$K\oplus H$ . Define maps
: $(\sigma\oplus\pi)(d)arrow B(K\oplus H)$ by$\sigma$$\sigma\oplus\pi$$\mu_{n}$$\mu_{n}(\sigma(a)\oplus\pi(a))=\sigma(a)\oplus\Phi_{n}(a)$,$a\in$ .

is completely contractive.
Then
Also $\mu_{n}(\sigma(s)\oplus\pi(s)arrow\sigma(s)\oplus\pi(s))$ , for all $s\in S$ .
Also $\mu_{n}(\sigma(a)\oplus\pi(a)arrow\sigma(a)\oplus\pi(a))$ , for all $a\in$ .
$\mu_{n}$99Now,$\lim_{narrow\infty}\sup\Vert\Phi_{n}(a)-\pi(a)\Vert\leq\lim_{narrow\infty}sub\Vert\sigma(a)\oplus\Phi(a)-\sigma(a)\oplus\pi(a)\Vert$

$=$

sub

$\Vert\mu_{n}(\sigma(a)\oplus\pi(a))-\sigma(a)\oplus\pi(a)\Vert$

Therefore$\lim_{narrow\infty}\Vert\Phi_{n}(a)-\pi(a)\Vert=0$

Now: Let :
of -algebras and let :$s\in S$ . We have to show that$(iii)\Rightarrow(iv)$$arrow \mathscr{B}$$\theta$$C^{*}$$\Phi$

be an identity preserving homomorphism
be a UCP that satisfies $\Phi(\theta(s))=\theta(s)$ ,$arrow \mathscr{B}$$\Phi(\theta(a))=\theta(a)$Let$B_{0}$

be the separable$a\in$ .

$C^{*}$-algebra in$\mathscr{B}$2.

$\forall a\in$generated by$\theta()\cup\Phi(\theta())\cup\Phi^{2}(\theta())\cup\cdots$

It is clear that.
By considering a faithful representation of
on a separable Hilbert space$H$ , we may
assume that $B_{0}\subseteq B(H)$ . Let : $B(H)arrow B(H)$ is a UCP map
either. Hereince : $arrow B(H)$ is
a representation on $H$ , we must have$\Phi(B_{0})\subseteq B_{0}$$B_{0}$$\tilde{\Phi}$$\overline{\Phi}/B_{0}=\Phi$$\overline{\Phi}(\theta(s))=\theta(s),$$\forall s\in S.$$\Phi(\theta(a))=\overline{\Phi}(\theta(a))=\theta(a)$$S$$\theta$$\forall a\in A$.

Hence the proof.
Let : $arrow B(H)$ be a faithful representation of
on $B(H)$$H$
for some Hilbert space . Put $\mathscr{B}=B(H)$ . Consider the -algebras of
all bounded seqences
and
in
and
respectively. Let: $\pi(A)arrow B(H)$ be the sequence of all completely contractive maps
such that$(iv)\Rightarrow(i)$$\pi$$C^{*}$$l^{\infty}()$$\mathscr{B}$$l^{\infty}(\mathscr{B})$$\Phi_{n}$$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(s)-\pi(s))\Vert=0$$\forall s\in S$.

To show that$\lim_{narrow\infty}\Vert\Phi_{n}(\pi(a)-\pi(a)\Vert>0$$\forall a\in$.

Let$\tilde{\Phi}$:$l^{\infty}()arrow l^{\infty}(\mathscr{B})$be defined as$\tilde{\Phi}(a_{1}, a_{2}\ldots a_{n}, \ldots)=(\Phi_{1}(a_{1}), \Phi_{2}(a_{2}), \ldots)$$(a_{1}, a_{2}, \ldots, a_{n}, \ldots)\in l^{\alpha}()$First we show that$l^{\infty}()\otimes M_{n}(C)$$\tilde{\Phi}$

is completely contractive. It is quite easy to see thatcan be identified isometrically withsame way identify
with
Thus
can be regarded as a map from$l^{\infty}(\mathscr{B}\otimes M_{n}(C))$$\tilde{\Phi}^{(n)}$$M_{n}(C))$$n$.

, for each positive integer$n.\tilde{\Phi}^{(n)}$$l^{\alpha}(\otimes M_{n}(C))$$l^{\infty}(\mathscr{B})\otimes M_{n}(C)$

.

. The

to
is the map induced by for each$l^{\infty}(\otimes M_{n}(C))$$l^{\infty}(\mathscr{B}\otimes$$\tilde{\Phi}$

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M. N. N. Namboodiri

is contractive andis contractive since$\Phi^{(n)}(\pi(I))arrow\pi(I)$ as.
) denotes the ideal of all sequences in
Let $C_{0}()$ (respectively(respectively ) that converges to zero in norm. Consider the map
It is easy to see that$\Phi^{(n)}$$\tilde{\Phi}^{(n)}$$narrow\infty$$\mathscr{A}$$C_{0}(\mathscr{B})$$\mathscr{B}$$\tilde{\Phi}_{0}:\frac{l^{\infty}()}{C_{0}()}arrow\frac{l^{\infty}(\mathscr{B})}{C_{0}(\mathscr{B})}$defined by$\tilde{\Phi}_{0}(x+C_{0}())=\tilde{\Phi}(x)+C_{0}(\mathscr{B})$

Then

,

$x\in l^{\infty}()$is completely contractive. Consider the embedding
defined by$\tilde{\Phi}_{0}$$l^{\infty}(d)$$\theta$: $arrow$$\theta(a)=(a, a, \ldots)+C_{0}(d)$Therefore$\tilde{\Phi}_{0}(\theta(s))=(\Phi_{1}(s), \Phi_{2}(s), \ldots)+C_{0}()$$=(s, s, \ldots, s, \ldots)+C_{0}(d)$$\forall s\in S$$=\theta(s)$Thus$\tilde{\Phi}_{0}:\frac{l^{\infty}(d)}{C_{0}()}arrow\frac{l^{\infty}(\mathscr{B})}{C_{0}(\mathscr{B})}$

such that$\tilde{\Phi}_{0}(\theta(s))=\theta(s)$$s\in S$.

$\Rightarrow\tilde{\Phi}_{0}$is a UCP since identity $1_{d}\in S$ . This is because, if:
and ifandalgebras with identities
linear map such that
$1_{d}$$\Vert\Phi(1_{d})\Vert=\Vert\Phi\Vert$then$\Phi$is positivity preserving. Then$darrow \mathscr{B}$$\Phi$$1_{\ovalbox{\tt\small REJECT}}$and$\mathscr{B}$are$C^{*}-$

is a contractive,$\tilde{\Phi}_{0}(\theta(a))=\theta(a)$for all $a\in$ . That

is,$(\Phi_{1}(a), \Phi_{2}(a), \ldots)+C_{0}(d)$$=(a, a, \ldots)+C_{0}()$$\Rightarrow\Vert\Phi_{n}(a)-a\Vertarrow 0$$a\in$

as $narrow\infty$$\square$2.20. Remarks. The above theorem is a noncommutative analogue of Wulberts theorem for hyperrigidity in function spaces such as $C(X)$ . This isbecause every contractive linear map on $C(X)$ is completely contractive
algebras is[3, 4]. So most of Arveson’s theorem for hyperrigidity for
proof.
ofwithout
these
some
state
We
valid for strong hyperrigidity also.
$C^{*}$

2.21. Corollary. Let be a strongly hyperrigid separable operator system,$S$

let $K$ be an ideal in and let $a\in\mapsto\dot{a}\in$with generated -algebra$d/K$ be the quotient map. Then is a strongly hyperrigid operator system
in $/K$ .
$C^{*}$$d_{1}$$\dot{S}$101
2.22. Theorem. Let $x\in B(H)$ be a self adjoint operator with atleast 3
points in its spectmm and let
be the -algebra generated by and 1.
$C^{*}$$x$Then(i) $G=\{1, x, x^{2}\}$ is a strongly hypeiTigid operator system for(ii) $G_{0}=\{1, x\}$ is not a strongly hyperrigid generator for .Of.

2.23. Theorem. Let-algebra ,Of and let$\{u_{1}, u_{2}, \ldots u_{n}\}$, whilebe a set of isometrices that generate a$C^{*}$$G=\{u_{1}, u_{2}, \ldots u_{n}, u_{1}^{*}, u_{1}+u_{2}^{*}u_{2}+\cdots+u_{n}^{*}u_{n}\}$

Then

$G$

is a strongly hyperrigid generator for.

.

2.24. Corollary. The set $G=\{u_{1}, u_{2}, \ldots, u_{n}\}$ , where, ofgenerators of the Cuntz algebra is strongly hyperrigid.
We conclude this section by remarking that many more implication ofstrong hyperrigidity theorem’ are to be investigated. However such results
will appear elsewhere.
$\sum_{k=1}^{n}u_{k}u_{k}^{*}=I$$\theta_{n}$‘3. TYPE II KOROVKIN THEOREMS
Recall that this section deals with the size of a test set $H$ in a -algebra.Of generated by $H$ . In $C[a, b]$ , there is no test set containing only twoelements.
Observe that (i) of 2.22 is already known, where as (ii) does not seem toexist in this generality. Does this result have a noncommutative analogue?
$C^{*}$

3. 1. Question. Let $x\in B(H)$ . Let be the -algebra generated by ad
X. Then it is known that $\{I, x, x^{*}x+xx^{*}\}$ is hyperrigid in . If spectmm
has atleast 3 distinct points, then is it tme that $\{I, x\}$ is not a hyperrigid
generator of ?
The following simple modification of 2.22 is possible.
$I$$C^{*}$$\sigma(x)$3.2. Proposition. Let $x\in B(H)$ be normal. Then $G=\{1, x, x^{*}x\}$ is a
hyperrigid set of generators for ,Of $=C^{*}(x)$ . If
contains three distinct points
will not be aand , on some straight line, then 1,
hyperrigid generator for .
We provide the proof for the sake of completion.
$\sigma(x)$$\lambda_{1},$$\lambda_{2}$$\{$$\lambda_{3}$$x\}$Proof. Statement (i) is already known. Now we prove (ii) using Arveson’sargument. Let $S=$ span 1,
and letdenote the spectmm of . For$f\in C(\sigma(x))$ , let $\phi_{k}(f(x))=f(\lambda_{k}),$ $k=1,2,3$ .
Thenis a multiplicative positive linear functional of norm 1 which isan irreducible representation of on . But
$\{$

$x\}$

$\sigma(x)$$x$$\phi_{k}$$\mathbb{C}$$\phi_{k}(\lambda 1_{\alpha}+\mu_{x})=\lambda+\mu\phi_{k}(x)$$=\lambda+\mu\lambda_{k},$

But
and

$\lambda_{2}$$\lambda_{3}$$k=1,2,3$ .

is a convex combination (assume without loss of generality) of. Therefore $\lambda_{2}=t\lambda_{1}+(1-t)\lambda_{3},0<t<1$ .

$\lambda_{1}$102
M. N. N. Namboodiri

Therefore $\phi_{2}(\lambda 1_{d}+\mu x)=t\phi_{1}(\lambda 1_{d}+\mu x)+(1-t)\phi_{3}\lambda 1_{d}+\mu x)$ .
are twoThus the positive linear functional $\phi=t\phi_{1}+(1-\phi)\phi_{3}$ and.
Therefore the irreducible
different completely positive extensions ofrepresentation fails to have unique extension property therefore one is
not hyperrigid.
$\phi_{2}$$\phi_{2}/S$$x$$\phi_{2}$$\square$

We conclude this section by stating a problem of Arveson [6].

3.3. Question. Let $I=[a, b],$ : $Iarrow R$ and $A\in B(H)$ be selfadjoint. Is$[1, A, f(A)]$ hyperrigid in $C^{*}(A)$ ? Arveson observes that in casehas discrete spectmm in $[a, b]$ and if is either strictly convex or strictly concave,
the answer is affirmative.
$f$

$A$

$f$

4. TYPE III KOROVKIN THEOREMS
is hyperrigid
Recall the classical Korovkin theorem says thatsystem. It would
is a
in $C[a, b]$ exactly when spanev
be interesting to examine its non commutative counterpart using
systems in-algebra. First we recall the notion of Ceby\v{s}ev system in
Banach spaces.
$\{f_{1}, f_{2}, f_{3}\}$$\{f_{1}, f_{2}, f_{3}\}$$\check{C}eby\check{s}ev$$\check{C}eby\check{s}$$\vee$$C^{*}$

4.1. Definition. Let $M$ be a subspace of a Banach space. $N$ is called asystem if each vector $N$ admits a unique closest point in $M$ .
A. Haar in 1918 [20] obtained the following characterization of finite
subspaces of $C(X),$ $X$ compact and Hausdorff. For
dimensional
algebras the study was camied out by A. G. Robertson, David Yost and
G. K. Pederson [16]$\check{C}eby\check{s}ev$$\check{C}eby\check{s}ev$$C^{*}$

4.2. Proposition. [7] Let $M$ be an n-dimensional subspace of $C(X)$ . Then
$M$ is a
system if and only if no non zero function in $M$ has morethat $n-1$ zeros.
$\check{C}eby\check{s}ev$

4.3. Theorem. [7] Let $X$ denote an interval $[a, b]$ or the unit circle . Theneachsystem $S=\{g_{0}, g_{1}, \ldots, g_{m}\}m\geq 2$ is a Korovkin set (hy$T$$\check{C}eby\check{s}ev$

pernigid)
It is to be remarked that finite Korovkin sets have been studied for func-tion spaces, commutative Banach algebras and for some special types ofalgebras [1]. We pose the following problem whose answer is not yet
known.
$C^{*}$

4.4. Question. Let $M$ be an dimensional subspace ofalgebras ,where $n\geq 3$ . Is it tme that $M$ is hyperrigid if it is a
subspace?
We conclude this section by mentioning few things regarding weak Ko$n$$C^{*}$$\check{C}eby\check{s}ev$

rovkin type theorems.
When approximation in the weak sense by completely positive linear
maps on $B(H)$ is considered, Korovkin type results have been obtainedin [13]. For example, recall the definition of weak Korovkin set introducedin [13]. It is as follows:

103
4.5. Definition. A subset

of $B(H)$ is called a weak Korovkin set if for, the relationeach netof completely positive maps satisfying
weakly, $s\in S$ implies
weakly $T\in B(H)$ .
One of the main theorems proved in [13] is as follows.
$S$$\Phi_{\alpha}(I)\leq I$$\Phi_{\alpha}$$\Phi_{\alpha}(s)arrow s$$\Phi_{\alpha}(T)arrow T$

be an irreducible set $B(H)$ such that contains the$C^{*}(S)$identity operator andcontains a non zero compact operator. Then
is a weak Korovkin set in $B(H)$ if and only if id has a unique completely
positive linear extension to $C^{*}(S)$ namely id.

4.6. Theorem. Let

$S$

$S$

$S$

$I$

$|s$

$|_{C^{*}(S)}$4.7. Remarks. The condition ‘id has a unique completely positive linis
ear extension to $C^{*}(S)$ means that the identity representation ofboundaly representation for in the sense of Arveson.
The following boundary theorem of Arveson enables to identify a number$|s$’$C^{*}(S)$$S$

of weak Korovkin sets.

4.8. Boundary theorem of Arveson. Let

be a irreducible set in $B(H)$such that contains the identity operator and $C^{*}(S)$ contains a non zerocompact operator. Then the identity representation of $C^{*}(S)$ is a boundary representation for , if and only if the quotient map : $B(H)arrow$$B(H)/K(H)$ is not completely isometric on span $(S+S^{*})$ where $K(H)$denote the set of all compact operators on $H$ .
One of the implications provides the following example.
$S$

$S$

$S$

$q$

4.9. Example. Let be an irreducible operator which is almost normal but
not normal, then the set $S=\{I, s, s^{*}s+ss^{*}\}$ is a weak Korovkin set.
$S$

Acknowledgement. The author is thankful to the organisers of the RIMS
Kyoto Symposium, Oct.27-29, 2010 for the invitation as well as for local
hospitality; CUSAT and NBHM-DAE, Govt. of India for financial support.
Also the author is thankful to Prof. B. V. Limaye for suggesting some modifications and corrections.
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$C^{*}$$C^{*}$