New tensor products of C*-algebras and characterization of type I C*-algebras as rigidly symmetric C*-algebras
Authors: Lee Hun Hee , Samei Ebrahim , Wiersma Matthew 
Published: February 2024

Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors

(
A
,
B
)
↦
A

⊗

α

B

(A,B)\mapsto A\otimes _{\alpha } B

, where

A

⊗
α

B

A\otimes _\alpha B

is a cross norm completion of

A
⊙
B

A\odot B

for each pair of C*-algebras

A
A

and

B
B

. For the first class of bifunctors considered

(
A
,
B
)
↦
A

⊗
p

B

(A,B)\mapsto A{\otimes _p} B

(

1
≤
p
≤
∞

1\leq p\leq \infty

),

A

⊗
p

B

A{\otimes _p} B

is a Banach algebra cross-norm completion of

A
⊙
B

A\odot B

constructed in a fashion similar to

p
p

-pseudofunctions

PF
p
∗

(
G
)

\text {PF}^*_p(G)

of a locally compact group. Taking a cue from the recently introduced symmetrized

p
p

-pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider

⊗

p
,
q

{\otimes _{p,q}}

for Hölder conjugate

p
,
q
∈
[
1
,
∞
]

p,q\in [1,\infty ]

– a Banach

∗
*

-algebra analogue of the tensor product

⊗

p
,
q

{\otimes _{p,q}}

. By taking enveloping C*-algebras of

A

⊗

p
,
q

B

A{\otimes _{p,q}} B

, we arrive at a third bifunctor

(
A
,
B
)
↦
A

⊗

C

p
,
q

∗

B

(A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B

where the resulting algebra

A

⊗

C

p
,
q

∗

B

A{\otimes _{\mathrm C^*_{p,q}}} B

is a C*-algebra.

For

G
1

G_1

and

G
2

G_2

belonging to a large class of discrete groups, we show that the tensor products
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Subscript normal r Superscript asterisk Baseline left-parenthesis upper G 1 right-parenthesis circled-times Subscript normal upper C Sub Subscript p comma q Sub Superscript asterisk Baseline normal upper C Subscript normal r Superscript asterisk Baseline left-parenthesis upper G 2 right-parenthesis">

C

r

∗

(

G
1

)

⊗

C

p
,
q

∗

C

r

∗

(

G
2

)

\mathrm C^*_{\mathrm r}(G_1){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G_2)

coincide with a Brown-Guentner type C*-completion of

ℓ

1

(

G
1

×

G
2

)

\mathrm \ell ^1(G_1\times G_2)

and conclude that if

2
≤

p
′

>
p
≤
∞

2\leq p’>p\leq \infty

, then the canonical quotient map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Subscript normal r Superscript asterisk Baseline left-parenthesis upper G right-parenthesis circled-times Subscript normal upper C Sub Subscript p comma q Sub Superscript asterisk Baseline normal upper C Subscript normal r Superscript asterisk Baseline left-parenthesis upper G right-parenthesis right-arrow normal upper C Subscript normal r Superscript asterisk Baseline left-parenthesis upper G right-parenthesis circled-times Subscript normal upper C Sub Subscript p comma q Sub Superscript asterisk Baseline normal upper C Subscript normal r Superscript asterisk Baseline left-parenthesis upper G right-parenthesis">

C

r

∗

(
G
)

⊗

C

p
,
q

∗

C

r

∗

(
G
)
→

C

r

∗

(
G
)

⊗

C

p
,
q

∗

C

r

∗

(
G
)

\mathrm C^*_{\mathrm r}(G){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G)\to \mathrm C^*_{\mathrm r}(G){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G)

is not injective for a large class of non-amenable discrete groups possessing both the rapid decay property and Haagerup’s approximation property.

A Banach

∗
*

-algebra

A
A

is symmetric if the spectrum

S
p

A

(

a
∗

a
)

\mathrm {Sp}_A(a^*a)

is contained in

[
0
,
∞
)

[0,\infty )

for every

a
∈
A

a\in A

, and rigidly symmetric if

A

⊗

γ

B

A\otimes _{\gamma } B

is symmetric for every C*-algebra

B
B

. A theorem of Kügler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kügler’s theorem by showing for C*-algebras

A
A

and

B
B

that

A

⊗

γ

B

A\otimes _{\gamma }B

is symmetric if and only if

A
A

or

B
B

is type I. In particular, a C*-algebra is rigidly symmetric if and only if it is type I. This strongly settles a question of Leptin and Poguntke from 1979 [J. Functional Analysis 33 (1979), pp. 119—134] and corrects an error in the literature.