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
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
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.