数理解析研究所講究録
1077 巻 1999 年 1-12

1

STRUCTURE OF THE GROUP OF
AUTOMORPHISMS OF -ALGEBRAS$\mathrm{C}^{*}$

京大数理研川村勝紀 (Katsunori Kawamura) 1

1

Introductionof the functional representation theorem
A non-commutative
for commutative unital -algebras was introduced in [2]. This generalization
was established via a non-commutative Gelfand transform mapping an unital(for some non-commutative product)-algebraA to an algebra ofon the set of pure states of $A$ viewed as a uniform K\"ahler bundle over thecan be seen as
spectrum of $A$ (See Sect. 3). The K\"ahler structure
a geometrical counterpart of Shultz’ characterization [5] of the set of pure.
states of a unital-algebras can be translated into
As a consequence, any statement
an equivalent statement in terms of uniform K\"ahler bundles. For example,
between two -algebras $A$ and $A$’ is in
the set of
between thecorrespondence with the set of uniform K\"ahler
K\"ahler bundles associated with $A$ and $A’[2]$ .
between -algebras and K\"ahler geWe think that this
exploited to get new insights in some problems
ometry can be
occurring in -algebras theory. Also, the non-commutative structure on the$\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{Z}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$$\mathrm{C}^{*}$$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}$$\mathrm{C}^{*}$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{o}1_{\mathrm{V}\mathrm{e}\mathrm{d}}$$\mathrm{C}^{*}- \mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}$$\mathrm{a}\mathrm{b}_{\mathrm{o}\mathrm{u}}\mathrm{t}$$\mathrm{c}^{*}$$\mathrm{C}^{*}$$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}\mathrm{o}^{-\mathrm{o}\mathrm{n}\mathrm{e}}$$*- \mathrm{i}\mathrm{S}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{S}$$\mathrm{i}_{\mathrm{S}\mathrm{o}\mathrm{m}}\mathrm{Q}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{S}\mathrm{m}\mathrm{s}$$\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}_{0}\mathrm{r}\mathrm{m}$$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{P}^{\mathrm{o}\mathrm{n}}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$$\mathrm{a}\mathrm{d}_{\mathrm{V}\mathrm{a}\mathrm{n}}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{u}\mathrm{S}\mathrm{l}\mathrm{y}$$\mathrm{C}^{*}$

le-mail: kaWamura@kurims.kyoto-u.ac.ip$\mathrm{C}^{*}$

2
space of functions on the set of pure states seems to be related with deformation quantization of Poisson manifolds and a better understanding of thislink might result in a fruitful interaction between these fields.
In this paper, by using non-commutative functional representation theorem, we study the structure of the group of automorphisms of -algebras interms of geometry of uniform K\"ahler bundles.
The paper is organized as follows.
section 2, we state our main theorem.
section 3, we review the theory of uniform K\"ahler bundle [2]. In section 4,
we introduce the orbit spectrum of a -algebra $A$ . It is the space of orbits
in the spectrum of $A$ of the group of automorphisms of $A$ . We decompose
the uniform K\"ahler bundle associated to $A$ by the orbit spectrum.
section5, we prove the main theorem.
$\mathrm{C}^{*}$$\cdot$

$\ln$

$\ln$

$\mathrm{C}^{*}$$\ln$

Structure of the group of automorphisms

2

We first state our main theorem using the following notation:
Let $A$ be a unital -algebra. Aut $A$ is the group of -automorphisms of $A$ ,$B$
is the spectrum of $A$ defined as the set of all the equivalence classes ofirreducible representations of $A$ , and is the set of pure states of $A$ . With
respect to the weak* topology,
is a uniform space [2], [1]. Since
acts on $B$ naturally, we define the orbit space A
denoting the
projection
by :corresponding natural.
$*$$\mathrm{C}^{*}$$\mathcal{P}$$\prime p$$\mathrm{A}\mathrm{u}\mathrm{t}A$$\equiv B/\mathrm{A}\mathrm{u}\mathrm{t}A$$p’$$Barrow\Lambda$

Theorem 2.1 There is an injective homomorphism
$\pi$

:

$\mathrm{A}\mathrm{u}\mathrm{t}A^{\mathrm{c}arrow}PU(\mathcal{P})\lambda\delta S(B)\Lambda$

where

PU
$S(B)^{\Lambda}\equiv${ :$\phi$$Barrow B:\emptyset$$(P) \equiv\prod_{b\in B}PU_{b}$,is a bijection such that$p’\mathrm{o}\phi=p’$},

is the projective unitary group on the representation space of the representative element in the spectrum $B$ and is the right action of
onPU
defined by$PU_{b}$$b$$S(B)^{\Lambda}$$\delta$$(\mathcal{P})$$\{u_{b}\}\delta_{\phi}\equiv\{u\phi-1(b)\}$for$\{u_{b}\}\in PU(\mathcal{P})_{r}u_{b}\in PU_{b}$and$\phi\in S(B)^{\Lambda}$.

3
The image of Aut $A$ under
byonPU$(P)\rangle\triangleleft sS(B)^{\Lambda}$$\pi$

is given in terms of a faithful action$\kappa$of$\mathcal{P}_{f}${ $g\in PU(P)\chi sS(B)\Lambda$ :$\kappa_{g}$is acting as a uniform homeomorphism on$\mathcal{P}$}.

By this theorem, we characterize an element of image of Aut $A$ underwhich is a uniform homeomorphism on $P$ .
as an element of PU
is an equivfor $[\pi]\in B$ where, defineFor
alence class of irreducible representations with the representative element$\pi$$(\mathcal{P})x\delta S(B)\Lambda$$\pi$$[\pi]$$\alpha[\pi]\equiv[\pi 0\alpha^{-1}]$$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}A$.

Corollary 2. In Theorem$\alpha b=b$
for any $b\in B$ } by is$\mathrm{I}$$\mathit{2}.\mathit{1}_{f}$the image of the subgroup {$\alpha\in$

Aut $A$ :$\pi${ $v\in PU(P):\kappa_{v}$ is a uniform homeomorphism on }$P$

$PU_{u}(\mathcal{P})\equiv$

where $PU_{u}(P)$ is identified with $PU_{u}(\mathcal{P})\cross\{1\}\subset PU(P)*ss(B)\Lambda$ .
Example 2.1 Let $X$ be a compact Hausdorff space. For the commutativeis isomorphic to the group Homeo $X$ of homeo-algebra $A\equiv C(X),$depends on the topological structure
morphisms on $X$ . So,.
is trivial. So, PU
of $X$ . Since$X$
Any compact Hausdorff space has uniformity and Homeo is equal to
the set of uniform homeomorphisms on $X[1]$ . The image of the injection ofis then equal to HomeoX.
Aut $A$ into$\mathrm{A}\mathrm{u}\mathrm{t}A$$\mathrm{C}^{*}$$\Lambda\equiv X/\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}x$$\mathcal{P}\cong X\cong B,$$(\mathcal{P})\rangle\triangleleft S(B)^{\Lambda}\cong s(B)^{\Lambda}$$PU(\mathcal{P}$$S(B_{A})^{\Lambda}$

By above argument, an element of”symmetry of a general noncommutative

is considered as a” topological-algebra $A$ .

$S(B_{A})^{\Lambda}$

Example 2.2 Let
is known that
Aut $A=$ {$\mathrm{A}\mathrm{d}U$:$U$$\mathcal{H}$$\mathrm{C}^{*}$

be a Hilbert space withis unitary on$\mathcal{H}$}$\dim \mathcal{H}\geq 1$$\cong PU(\mathcal{H})\equiv$

and$A\equiv \mathcal{L}(\mathcal{H})$. lt{projective unitaries on$\mathcal{H}$. Any automorphismby the isomorphism Aut$A$ and.
leaves unchanged the elements of the spectrum of
Therefore $S(B)^{\Lambda}=\{id_{B}\}$ . Therefore$A\ni \mathrm{A}\mathrm{d}U\vdasharrow[U]\in PU(\mathcal{H})$$\Lambda\equiv B/\mathrm{A}\mathrm{u}\mathrm{t}A\cong B$

PU$(\mathcal{P})\lambda S(B)^{\Lambda}=PU(\mathcal{P})\cross\{1\}$

.

}

4

In theorem 2.1, the image of{$\{[\pi_{b}(U)]\}_{bB}\in$

$\pi$

:

of
$U$

$\mathrm{A}\mathrm{u}\mathrm{t}A$isis unitary on$\mathcal{H}$}$\subset PU(\mathcal{P})$where the irreducible representationof $A$ is the representative element of$b\in B$ and we denote $[U]\equiv\{e^{it}U:t\in \mathrm{R}\}$ for a unitary operator.
$\pi_{b}$$U$

Example 2.3 For a -algebra $A,$ is a primitive ideal of $A$ if there isan irreducible representation of $A$ such that
. The primitive$A$$A$
spectrum ofis the set of all primitive spectrums of . Assume $A$ isa simple-algebra. Then the primitive spectrum of $A$ consists of only
one point. By definition, the Jacobson topology of the spectrum $B$ is thetrivial topology, that is, the open sets of $B$ are the empty set and $B$ itself [4]. So Homeo $B=S(B)\equiv$ {permutation of $B$ }. Furthermore, if
A is 1-point (we call $A$ automorphic), then $S(B)^{\Lambda}=S(B)=$ Homeo $B$ .
Thus, PU
Homeo . Therefore
is a subgroup
$A$
of PU
Homeo if is simple and automorphic.
$\mathrm{C}^{*}$$\mathcal{I}$$\mathrm{k}\mathrm{e}\mathrm{r}\pi=\mathcal{I}$$\pi$$\mathrm{C}^{*}$$(\mathcal{P})\rangle\triangleleft S(B)^{\Lambda}=PU(\mathcal{P})\rangle\triangleleft$$\mathrm{C}^{*}$$\mathrm{A}\mathrm{u}\mathrm{t}A$$B$$(\mathcal{P})\lambda$

3$B$-geometryIn this section, we review the characterization of the set of pure states andthe spectrum of a -algebra following [2].
Let $(f, E, M)$ be a surjective map $f:Earrow M$ between two sets $E,$ $M$ .
$\mathrm{C}^{*}$

Definition 3.1 $(f, E, M)$ is a formal K\"ahler bundle if there is a familyof If\"ahler manifolds indexed by $M$ and $E= \bigcup_{m\in M}E_{m}$ and $f(x)=$$\{E_{m}\}_{m\in M}$

$m$

if

$x\in E_{m}$.

We simply denote $(f, E, M)$ by $E$ .
Assume now that $E$ and $M$ are topological spaces.

Definition 3.2 $(f, E, M)$ is called a uniform K\"ahler bundle if $(f, E, M)$ is
a formal K\"ahler
is open, continuous, the topology of is a uniform
topology and the relative topology of each fiber is equivalent to the K\"ahler
topology of its fiber.
$bund\iota_{e_{f}}f$$E$

For a uniform topology, see [1]. The weak*-topology on the set of purestates of -algebra is a uniform topology.
$\mathrm{C}^{*}$

5

Definition 3.3 Two formal K\"ahler bundle $(f, E, M)_{f}(f’, E’, M’)$ are isomorphic if there is a pairof bijections : $Earrow E’$ and : $Marrow M_{f}’$such that$(\beta, \phi)$$\phi$$\beta$$f’\mathrm{o}\beta=\phi \mathrm{o}f$$\beta$$E$$f$$\cong$$\downarrow$$E’$$\downarrow$$f’$$M\cong M’$$\phi$: $f^{-1}(m)arrow(f’)^{-1}(\emptyset(m))$ is a holomorphic
and any restriction
K\"ahler isometry for any $m\in M$ . We calla formal K\"ahler isomorphism
between $(f, E, M)$ and $(f^{J}, E’’, M)$ .
$\beta|_{f^{-1}(m)}$$(\beta, \phi)$

By definition of a formal K\"ahler bundle isomorphism
between
$(f’,
E”,
M),$
$(f, E, M)$ andis uniquely determined by : For $m\in M$ ,the value $\phi(m)$ is given by $\phi(m).=f’(\beta(e))$ with arbitrary $e\in f^{-1}(\{m\})$ .
$(\beta, \phi)$$\phi$$\beta$’Definition 3.4 Two uniform K\"ahler bundles $(f, E, M))(f’, E’, M)$ are isobetween $(f, E, M)$ andmorphic if there is a formal K\"ahler isomorphism$(f’, E^{l}, M’)$ such thatis a homeomorphisms; and is a uniform homeoa uniform K\"ahfer isomorphism between $(f, E, M)$morphism. We calland $(f’, E^{J}, M)’$ .
$(\beta, \phi)$$\phi$$\beta$$(\beta, \phi)$

By definition, any uniform K\"ahler bundle is a formal K\"ahler bundle. Fora uniform K\"ahler bundle $E$ , we define:

Definition 3.5$1_{\mathrm{S}\mathrm{O}}\overline{(}E)\equiv$$1\mathrm{s}\mathrm{o}(E)\equiv$

the group of formal K\"ahler bundle isomorphisms of $E$ ,the group of uniform K\"ahler bundle isomorphisms of .
$E$

from
By the GNS representation, there is a natural projection$B$of pure states onto the spectrum . We consider $(p,P, B)$ as a
the setmap of topological spaces where $P$ is endowed with weak* topology and $B$is endowed with the Jacobson topology.
In Ref.[2], the following results are proved.
$p:\mathcal{P}arrow B$$\mathcal{P}$6

Theorem 3.1 (Reduced atomic realization) For any unital
is a uniform K\"ahler bundle.

$C^{*}$-algebra $A$ ,$(p,\mathcal{P}, B)$, let
be some irreducible representation
For a fiber, correspondbelonging to $b\in B$ . Towithdenoting a vector state.
where
has a K\"ahler manifold structure induced by this correspondence
Then.
from projective Hilbert space$\mathcal{P}_{b}\equiv p^{-1}(b)$$(\pi_{b}, \mathcal{H}_{b})$$[x_{\rho}]\in \mathcal{P}(\mathcal{H}_{b})\equiv(\mathcal{H}_{b}\backslash \{0\})/\mathrm{C}^{\mathrm{x}}$$\rho\in \mathcal{P}_{b}$$\rho=\omega_{x_{\rho}}0\pi_{b}$$\omega_{x_{\rho}}=<x_{\rho}|(\cdot)x_{\rho}>$$\omega_{x_{\rho}}$$\mathcal{P}_{b}$$\mathcal{P}(\mathcal{H}_{b})$Theorem 3.2 Let
dles
and

$A_{i}$$(p_{i},\mathcal{P}_{i}, B_{i})_{f}i=1,2$$(p_{1},\mathcal{P}_{1,1}B)$-algebras with associated uniform K\"ahler bun. Thenandare *-isomorphic if and only if
are isomorphic as uniform K\"ahler bundles.

be$C^{*}$$A_{2}$$A_{1}$$(p_{2},\mathcal{P}_{2,2}B)$be the group $of*$ -automorphisms of a -algebra
Corollary 3.1 Let
$A$
, and
with an associated uniform K\"ahler bundle
be thegroup of uniform K\"ahler bundle automorphisms on P. Then there is a groupisomorphism
Aut.
$C^{*}$$\mathrm{A}\mathrm{u}\mathrm{t}A$$\mathcal{P}=(p, \mathcal{P}, B)$$\mathrm{I}\mathrm{s}\mathrm{o}\mathcal{P}$$A\cong 1\mathrm{s}\mathrm{o}\mathcal{P}$, let:
and inducedFor. Then: $Barrow B$ defined bybecomesbijectiona uniform K\"ahler bundle automorphism of .
We call these objects -geometry since any -algebra can be reconstructed from the associated uniform K\"ahler bundle [2] and, therefore, any-algebra is determined by such a geometry.
in the language
By the above result, we can consider the structure of.
of$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}A$$\beta_{\alpha}\equiv\alpha^{*}|_{P}$$\mathcal{P}arrow \mathcal{P},$$\alpha^{*}(\rho)\equiv\rho 0\alpha^{-1}$$\phi_{\alpha}([\pi_{)}])\equiv[\pi 0\alpha^{-1}]$$\phi_{\alpha}$$(\beta_{\alpha}, \phi_{\alpha})$$\mathcal{P}$$C^{*}$$\mathrm{C}^{*}$$\mathrm{C}^{*}$$\mathrm{A}\mathrm{u}\mathrm{t}A$$\mathrm{I}\mathrm{s}\mathrm{o}\mathcal{P}$4

Orbit decomposition of a K\"ahler bundle

We decompose the set of pure states and the spectrum of a-algebra $A$
as a uniform K\"ahler bundle. By using this decomposition, we describe automorphisms of $A$ in each decomposed component in the next section.
of a -algebra $A$ are called
In $[3](11, \mathrm{p}906)$ , two pure states andautomorphic if there is an automorphism of $A$ such that $\rho’=\rho 0\alpha$ . For
example, any two pure states of a uniform matricial algebra are automorphic
general, the set of pure states of a -algebra([3] , Theorem 12.3.4).
$\mathrm{C}^{*}$$\rho’$$\mathrm{C}^{*}$$\rho$$\alpha$$1\mathrm{I}$$\ln$$\mathrm{C}^{*}$

7is divided into a disjoint union of automorphic component. Therefore, eachic components.
automorphism induces transformations on each
The idea on which this section is based comes from this point of view.
. is naturally acting on the spectrum $B$ by
Let
for $g\in G$ and $[\pi]\in B$ . So, we define the space A of orbits of in $B$ ,$\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\dot{\mathrm{h}}$$G\equiv \mathrm{A}\mathrm{u}\mathrm{t}A$$g[\pi]\equiv[\pi \mathrm{o}g^{-1}]$

$G$

$G$

$\Lambda\equiv B/G$

and call it the orbit spectrum.
Lemma 4.1 The orbit of

$G$

$G$

in

,

acts naturally also on$\mathcal{P}$

and

$B$

are in$one-t_{\mathit{0}}$$\mathcal{P}$by$g\rho\equiv\rho \mathrm{o}g^{-1}$.

-one correspondence.

. We define
be an orbit of throughis the unitary equivalence class of irreducible representations of $A$
where
with a representative element , given by the GNS representation of . For
by uniquenessis unitarily equivalent toof the GNS representation. Then the map : $P/Garrow\Lambda$ is well defined. By
definition, maps orbits in to orbits in . And $\Psi(c_{\rho})=Gp(\rho)$ . Hence,is onto.
are GNS represenand
If $\Psi(G\rho)=\Psi(G\rho’)$ and,. Since two automorphic
tations of $\rho’\in P$ respectively, then,pure states have GNS representation spaces with the same dimension, there
which acts on
are $g\in G$ , a representative element,. By irreducibility
such thatand a unitary operator onof , we can choose a unitary element in $A$ such that

Proof. Let$G$$G\rho$$\Psi(G\rho)\equiv G[\pi_{\rho}]\in\Lambda$$\rho\in P$$[\pi_{\rho}]$$\rho$$\pi_{\rho}$$g\rho\in G\rho,$$g\pi_{\rho}\equiv\pi_{\rho}\mathrm{o}g^{-1}\in G[\pi_{\rho}]$$\pi_{g\rho}$$\Psi$$\Psi$$B$$\mathcal{P}$$\Psi$$(\pi_{\rho}, \mathcal{H}_{\rho}, X_{\rho})$$(\pi_{\rho’’\rho’\rho}\mathcal{H}\prime X’)$

$G[\pi_{\rho}]=G[\pi_{\rho’}]$$\rho,$$\pi’\in[\pi_{\rho’}]$$U$$\mathcal{H}_{\rho},$$\rho’=\omega_{x}0\pi’\rho$$\pi=\mathrm{A}\mathrm{d}U\mathrm{o}g\pi_{\rho}$$\mathcal{H}_{\rho}$$V$$\pi_{\rho}$$\rho’=(g\rho)\mathrm{o}\mathrm{A}\mathrm{d}V=(\mathrm{A}\mathrm{d}V*\mathrm{o}g)\rho\in G\rho$(see [3], II, 10.2.6.). Therefore $G\rho=G\rho’$ .

$\Psi$

is an injection.

$arrow p’$$arrow p$$\mathcal{P}$$\downarrow$$B/G=\Lambda$$B$$\nearrow\Psi$$\mathcal{P}/G$

We decompose$\mathcal{P}$by A into the family of uniform K\"ahler bundles.

1

8

:

be the natural projection with fibers given
A. Letby. By lemma 4.1,
andin $B$ and , respectively. Let
are orbits offorA.
K\"ahlerfor eachThenbecomes a uniform
bundle with therelative topology such that its total space
is automorphic, that is, any
are transformed by some automorphisms of $A$ each other.
two elements of
We obtain a decomposition

Let

$p’$

$Barrow\Lambda=B/\mathrm{A}\mathrm{u}\mathrm{t}A$$B^{\lambda}\equiv(p’)^{-1}(\lambda),$$\lambda\in$$G$$\mathcal{P}^{\lambda}$$B^{\lambda}$$\mathcal{P}^{\lambda}\equiv\bigcup_{b\in B^{\lambda}}\mathcal{P}_{b}$$p^{\lambda}\equiv p|_{P^{\lambda}}$$\mathcal{P}$$\lambda\in$$\lambda\in\Lambda$$(p^{\lambda}, \mathcal{P}^{\lambda}, B^{\lambda})$$\mathcal{P}^{\lambda}$$\mathcal{P}^{\lambda}$$(p,P, B)= \bigcup_{\lambda\in\Lambda}(p^{\lambda}, \mathcal{P}^{\lambda\lambda}, B)$of a uniform K\"ahler bundle.
in the same orbit
Any two elements
have representative representation spaces with the same dimensions. For an orbit A, letbea Hilbert space corresponding to a representative element of some point in. We can choose a representative element belonging toan orbitwhichacts on the same Hilbert space
.
Let
and
:
defined by $p^{o}(\xi, b)=b$ for.
andbecomes a formal K\"ahler
with
fiber
for
.
$b,$

$b’$

$B^{\lambda}$$\in\Lambda$$\mathcal{H}_{\lambda}$$B^{\lambda}$$B^{\lambda}$$\mathcal{H}_{\lambda}$$\mathcal{P}^{o}\equiv\bigcup_{\lambda\in\Lambda}\mathcal{P}(\mathcal{H}_{\lambda})\cross B^{\lambda}$$b\in B^{\lambda}$$\xi\in \mathcal{P}(\mathcal{H}_{\lambda})$$\mathcal{P}(\mathcal{H}_{\lambda})\cross\{b\}$$p^{o}$$’\rho\circarrow B$.ndle$(p^{O}, \mathcal{P}^{O}, B)$$\mathrm{b}\mathrm{u}$$b\in B^{\lambda}\subset B$

Theorem 4.1 $(p, P, B)$ and $(p^{O}, P^{o}, B)$ are isomorphic as formal K\"ahler
bundles.

Proof. Fix a family of representative elements. Define :on
ifby, whereif
andthe identity map on . Thenis fiber-wise holomorphic isometry. Then,
isomorphism between
and$b\in B^{\lambda}$$\mathcal{H}_{\lambda}$$\beta$$[\pi_{\rho}]=b\in B^{\lambda}$$\mathcal{P}arrow \mathcal{P}^{0}$$\rho=\omega_{x_{\rho}}0\pi_{b}$

$B$

of

$\{\pi_{b}\}_{b\in B}$

such that$\pi_{b}$acts$\beta(\rho)\equiv([x_{\rho}], b)\in \mathcal{P}(\mathcal{H}_{\lambda})\cross B^{\lambda}$. Let: $Barrow B$ be
for
. And
becomes a formal K\"ahler$x_{\rho}\in \mathcal{H}_{\lambda}$$\beta(\mathcal{P}_{b})=P(\mathcal{H}_{\lambda})\cross\{b\}$$(p,\mathcal{P}, B)$$B$$(\beta, \phi)$$(p^{O}, \mathcal{P}^{O}, B)$.

$\phi$$b\in B^{\lambda}$$\beta$1Corollary 4.1 (Orbit decomposition) Let $\mathcal{P}=(p,P, B)$ be a uniform K\"ahler
bundle associated with a -algebra. Then there is a uniform K\"ahler bundle
withsuch that.
$\sigma$$\mathcal{P}^{o}=(p^{O}, P^{O}, B)$

Proof.

$\mathcal{P}\cong\prime P^{O}$$P^{o}= \bigcup_{\lambda\in\Lambda}\mathcal{P}(\mathcal{H}_{\lambda})\cross B^{\lambda}$

the previous theorem, let the topology ofbe the induced topol$P$
becomes a uniform K\"ahler bundle which isogy from . Then.
isomorphic to1$\ln$$\mathcal{P}^{o}$$(p^{o\prime p\circ},, B)$$(p, \mathcal{P}, B)$95

Proof of the main theorem

assoBy Corollary 4.1, we identify the uniform K\"ahler bundle
corresponding tociated with a -algebra $A$ and its orbit decomposition
be the group of -automorphisms of athe orbit spectrum A. Let
algebra $A$ with associated uniform K\"ahler bundle $’\rho_{=}(p,P, B)$ and the orbit
with its orbit decomposition .
spectrum A. By corollary 4.1, weandare defined in Theorem 2.1 andRecall that PU
Definition 3.5.
respectively on the set of pureof PU
We define actions
states by$\mathcal{P}=(p, \mathcal{P}, B)$$\mathcal{P}^{o}$$\mathrm{C}^{*}$$*$$\mathrm{A}\mathrm{u}\mathrm{t}A$$\mathrm{C}^{*}-$$\mathcal{P}^{o}$$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\underline{\mathrm{i}\mathrm{f}\mathrm{y}}P$$S(B)^{\Lambda}$$(\mathcal{P}),$$t,$$1\mathrm{s}\mathrm{o}\mathcal{P}$$S(B)^{\Lambda}$$(\mathcal{P}),$$s$$t_{u}(\xi, b)$$\equiv$$s_{\phi}(\xi, b)$$\equiv$$(u_{b}\xi ’, \phi(b)b ))$$($for $u=\{u_{b}\}_{b\in B}\in PU(P),$
we define injective homomorphisms
, by

$\xi$

$\phi\in S(B)^{\Lambda}$$\tau$

andand,$\mathcal{P}$$(\xi, b)\in \mathcal{P}^{o}$$\sigma$of PU. With these actions,$(P)$and of$S(B)^{\Lambda}$

into$\mathrm{I}\overline{\mathrm{s}\mathrm{o}}P$$\equiv$$\tau_{u}$$($$t_{u}$’$( s_{\phi} ,’ id_{B}\phi )),$$\equiv$$\sigma_{\phi}$

respectively, for$u\in PU(\mathcal{P}),$$\phi\in S(B)^{\Lambda}$.

Lemma 5.1$\tau(PU(\mathcal{P}))=\{(\beta, \phi)\in 1\overline{\mathrm{s}\mathrm{o}}p:\emptyset=idB\}\equiv G_{3}$.

is contained in . On the other hand,
becomes a holomorphic K\"ahler isometry on
for any $g=(\beta, id_{B})\in G_{3},$each fiber by definition of . So it becomes a projective unitary on eachfiber by Wigner’s theorem. Thus, there is a family of projective unitaries
corresponding to .
1
,
,
where
is
the
we
obtainandFordefined in Theorem 2.1. From this follows
on PU
right action of
the relation.
(Eq.5.1)

Proof. By definition of$\tau,$$G_{3}$$\tau(PU(\mathcal{P}))$$\beta$$G_{3}$$g$$u\in PU(\mathcal{P})$$S(B)^{\Lambda}$$\phi\in S(B)^{\Lambda}$$\delta$$s_{\phi}t_{u^{S}\emptyset}-1=t_{u\mathit{5}_{\phi}}$$(\mathcal{P})$$\sigma_{\phi^{\mathcal{T}_{u}\sigma}\emptyset^{-1}}=\tau_{u\mathit{5}_{\phi}}$

. Let
on
be
of
Consider the action. We obtain the following
andthe group generated byisomorphism between semi-direct products of groups.
$\tilde{\delta}_{a}=\mathrm{A}\mathrm{d}_{a}$$\tau(PU(\mathcal{P}))$$a\in\sigma(S(B)^{\Lambda})$$\sigma(S(B)^{\Lambda})$$\tau(PU(\mathcal{P}))$$G_{2}$

10

Lemma 5.2
PU$(\mathcal{P})\lambda_{\delta}S(B)^{\Lambda}\cong\tau(PU(\mathcal{P}))\lambda\overline{s}\sigma(S(B)^{\Lambda})=G_{2}$.

Proof. By Eq.5.1 and the definition of semi-direct product, the lemma follows.
1

Proof of Theorem 2.1 (main Theorem). By lemma 5.2, PU

is embedded intoLet

$1\overline{\mathrm{s}\mathrm{o}}P$$(\mathcal{P})\lambda sS(B)\Lambda$

as a subgroup.
$\kappa$: PU(Eq.5.2)$(\mathcal{P})\lambda_{\mathit{5}}s(B)^{\Lambda}\simeq\Rightarrow \mathrm{I}\overline{\mathrm{s}\mathrm{o}}P$for
be defined by
On the other hand, by Corollary 3.1, there is an isomorphism$(u, \phi)\in PU(\mathcal{P})\lambda_{\delta}s(B)\Lambda$$\kappa(u, \phi)\equiv\tau_{u}\sigma_{\phi}=(t_{u}s_{\emptyset}, \emptyset)$:$\pi_{1}$We denotelemma 5.1, there is
Therefore,$\pi_{1}(\alpha)\equiv(\beta_{\alpha}, \phi_{\alpha})$$\mathrm{A}\mathrm{u}\mathrm{t}A\cong \mathrm{I}_{\mathrm{S}}\mathrm{o}\mathcal{P}\subset 1\overline{\mathrm{s}\mathrm{o}}\mathcal{P}$for. Then,$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}A$$u^{\alpha}\in PU(\mathcal{P})$.

such that.

(Eq.5.3)$(\beta_{\alpha^{\mathrm{O}S_{\phi_{\overline{\alpha}^{1}}}}}, idB)\in G_{3}$$\beta_{\alpha}\mathrm{o}s_{\phi\overline{\alpha}^{1}}=t_{u^{\alpha}}$$\pi_{1}(\alpha)=(\beta\alpha’\emptyset\alpha)=(t_{u}\alpha s\phi\alpha’\phi\alpha)=\tau_{u}\alpha\sigma\phi\alpha$

By this calculation and Eq.5.2, (Aut $A$ ). We denote an element of PU
Let
which satisfies the product law. So,$\beta_{\alpha}=t_{u^{\alpha}\phi_{\alpha}}s$$\pi\equiv\kappa^{-1}|c_{2}\mathrm{o}\pi_{1}$.

.

$\subset G_{2}=\kappa(PU(P)\rangle\triangleleft_{\delta}S(B)^{\Lambda})$$\pi_{1}$. By$(\mathcal{P})\lambda_{\mathit{5}}s(B)^{\Lambda}$.

by$u\cdot\delta_{\phi}$$(u\cdot\delta\phi)(u\cdot\delta_{\emptyset’});=\{u(u\delta\emptyset)’\}\cdot\delta\phi\phi’$for$u\cdot\delta_{\phi},$$u^{l}\cdot\delta_{\phi}’\in PU(\mathcal{P})\rangle\triangleleft_{\mathit{5}}S(B)^{\Lambda}),$$u,$ $u’\in PU(\mathcal{P})$and$\phi,$$\phi’\in S(B)^{\Lambda}$. Then$\pi(\alpha)=u^{\alpha}\cdot\delta_{\phi^{\alpha}}\in PU(’P)\rangle\triangleleft\delta s(B)^{\Lambda}$. Then we obtain the injective homomorphism for which weforhave been looking$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}A$$\pi$$\mathrm{A}\mathrm{u}\mathrm{t}A$$\pi_{1}$PU$\mathrm{c}arrow$$\nearrow$$\searrow$$G_{2}$$(\mathcal{P})\rangle\triangleleft sS(B)^{\Lambda}$$\kappa^{-1}|_{G_{2}}$.

11

By Corollary 3.1, we obtain$\pi(\mathrm{A}\mathrm{u}\mathrm{t}A)=(\kappa^{-1}|_{G_{2}1}0\pi)(\mathrm{A}\mathfrak{U}\mathrm{t}A)=\kappa^{-1}|_{G_{2}}(\mathrm{I}\mathrm{s}\mathrm{o}P)$$=\kappa^{-1}|_{G_{2}}$

$=$

{

(

$\{(\beta,$$\emptyset)\in \mathrm{I}\overline{\mathrm{s}\mathrm{o}}\mathcal{P}$$g\in PU(\mathcal{P})\rangle\triangleleft_{\delta}s(B)^{\Lambda}$::$\kappa_{g}$$\beta$(by equation Eq.5.3)

is a uniform homeomorphism on

acts on$\prime p$)

as a uniform homeomorphism},from which the statement of Theorem 2.1 inunediately follows.

6$\mathcal{P}\}$

1

Conclusion

In this paper, we have obtained the orbit decomposition of the uniform K\"ahler
bundles and the group of automorphisms.
The next step would be to consider the orbit decomposition of the algebra
this context, the meaning of decomposition has to be cleared. It
itself.
might be a decomposition like by crossed products, free products ofalgebras.
We are studying geometrical objects corresponding to modules, crossed
product, subalgebra, -homomorphism and etc are currently under study for
non-commutative -algebras. They are realization of non-commutative geometry by “ real” geometry defined as the set of points and its function space[2]. So, they must be direct generalization of the geometry of commutative
case leading to new geometrical structures for which we would like to give a
better understanding.
$\ln$$\mathrm{C}^{*}-$$*$$\mathrm{C}^{*}$

References[1] N.Bourbaki, Elements of Mathematics, General topology part I,
Addison-Wesley Publishing company 1966.

[2] R.Cirelli, A.Mani\‘a and L.Pizzocchero, A functional representation of
non-commutative -algebras, Rev.Math.Phys. Vol.6, No.5 (1994) 675$C^{*}$

697.
[3] R.V.Kadison and J.R.Ringrose, Fundamentals of the theory of operator
algebras I $\sim IV$, Academic Press (1983).

12[4] G.K.Pedersen,
Press (1979)$C^{*}$-algebras and their automorphism groups, Academic[5] F. W. Shultz, Pure states as a Dual object forMath. Phys. Vol. 82, 497-502 (1982).

$C^{*}$-algebras, Commun.