groupoid C*-convolution algebras

groupoid C*-convolution algebras

0.1

Introduction: Background and definition of the groupoid C*–convolution algebra

Jean Renault introduced in ref.

[

6

]

the

C

*

–

algebra

of a

locally compact groupoid

G

as follows: the space of

continuous functions

with compact support on a

groupoid

𝖦

is made into a *-algebra whose

multiplication

is the

convolution

, and that is also endowed with the smallest

C

*

–norm which makes its

representations

continuous

, as shown in ref.

[

3

]

. Furthermore, for this convolution to be defined, one needs also to have a

Haar system

(

http://planetmath.org/GroupoidRepresentationsInducedByMeasure

)
associated to the locally compact groupoids (

http://planetmath.org/LocallyCompactGroupoids

)

𝖦

that are then called

measured groupoids

because they are endowed with an associated Haar system which involves the

concept

of measure, as introduced in ref.

[

1

]

by P. Hahn.

With these concepts one can now sum up the definition (or construction) of the

groupoid

C

*

-convolution algebra

, or

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

groupoid

C

*

-algebra, as follows.

Definition 0.1

.

a

groupoid C*–convolution algebra

,

G

C

⁢

A

, is defined for

measured groupoids

as a

*–algebra with “

*

” being defined by convolution so that it has a smallest

C

*

–norm which makes its representations continuous

.

Remark 0.1

.

One can also produce a functorial construction of

G

C

⁢

A

that has additional interesting properties.

Next we recall a result due to P. Hahn

[

2

]

which shows how

groupoid representations

relate to
induced *-algebra representations and also how–under certain conditions– the former can be derived from
the appropriate *-algebra representations.

Theorem 0.1

.

(source: ref.

[

2

]

). Any representation of a groupoid

(

G

,

C

)

with Haar measure

(

ν

,

μ

)

in a separable Hilbert space

H

induces a *-algebra representation

f

↦

X

f

of the associated
groupoid algebra

Π

⁢

(

G

,

ν

)

in

L

2

⁢

(

U

G

,

μ

,

H

)

with the following properties:

(1) For any

l

,

m

∈

H

, one has that

|

<

X

f

(

u

↦

l

)

,

(

u

↦

m

)

>

|

≤

∥

f

l

∥

∥

l

∥

∥

m

∥

and

(2)

M

r

⁢

(

α

)

⁢

X

f

=

X

f

⁢

α

∘

r

, where

M

r

:

L

∞

(

U

𝖦

,

μ

⟶

L

[

L

2

(

U

𝖦

,

μ

,

ℋ

]

, with

M

r

⁢

(

α

)

⁢

j

=

α

⋅

j

.

Conversely

, any *- algebra representation with the above two properties induces a groupoid representation, X, as follows:

<

X

f

,

j

,

k

>

=

∫

f

(

x

)

[

X

(

x

)

j

(

d

(

x

)

)

,

k

(

r

(

x

)

)

d

ν

(

x

)

]

.

(viz. p. 50 of ref.

[

2

]

).

Furthermore, according to Seda (ref.

[

10

,

11

]

), the

continuity

of a Haar system is

equivalent

to the continuity of the

convolution product

f

*

g

for any pair

f

,

g

of continuous functions with compact support. One may thus conjecture that similar results could be obtained for

functions

with

locally compact

support

in dealing with convolution products of either locally compact groupoids or

quantum groupoids

. Seda’s result also implies that the convolution algebra

C

c

⁢

(

𝒢

)

of a groupoid

𝒢

is closed with respect to convolution if and only if the fixed Haar system associated with the measured groupoid

𝒢

is

continuous

(see ref.

[

3

]

).

Thus, in the case of groupoid algebras of

transitive groupoids

, it was shown in

[

3

]

that any representation of a measured groupoid

(

𝒢

,

[

∫

ν

u

⁢

𝑑

λ

~

⁢

(

u

)

]

=

[

λ

]

)

on a separable Hilbert space

ℋ

induces a

non-degenerate

*-representation

f

↦

X

f

of the associated groupoid algebra

Π

⁢

(

𝒢

,

ν

,

λ

~

)

with properties formally similar to (1) and (2) above.
Moreover, as in the case of groups,

there is a correspondence between the

unitary representations

of a groupoid and its associated C*-convolution algebra representations

(p. 182 of

[

3

]

), the latter involving however

fiber bundles

of

Hilbert spaces

instead of single Hilbert spaces.

References

1

P. Hahn: Haar measure for

measure groupoids

.,

Trans. Amer. Math. Soc

.

242

: 1–33(1978).

2

P. Hahn: The

regular representations

of measure groupoids.,

Trans. Amer. Math. Soc

.

242

:35–72(1978).

Theorem

3.4 on p. 50.

3

M. R. Buneci.

Groupoid Representations

, Ed. Mirton: Timishoara (2003).

4

M.R. Buneci. 2006.,

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

Groupoid C*-Algebras.,

Surveys in Mathematics and its Applications

, Volume 1: 71–98.

5

M. R. Buneci.

Isomorphic

groupoid C*-algebras associated with
different Haar systems.,

New York J. Math.

,

11

(2005):225–245.

6

J. Renault. A groupoid approach to C*-algebras,

Lecture Notes in Math

., 793, Springer,
Berlin, (1980).

7

J. Renault. 1997. The Fourier Algebra of a Measured Groupoid and Its Multipliers,

Journal of

Functional Analysis

,

145

, Number 2, April 1997, pp. 455–490.

8

A. K. Seda: Haar measures for groupoids,

Proc. Roy. Irish Acad.
Sect. A

76

No. 5, 25–36 (1976).

9

A. K. Seda: Banach bundles of continuous functions and an

integral

representation theorem,

Trans. Amer. Math. Soc.

270

No.1 : 327-332(1982).

10

A. K. Seda: On the Continuity of Haar measures on topological groupoids,

Proc. Amer Math. Soc.

96

: 115–120 (1986).

11

A. K. Seda. 2008.

Personal communication

, and also Seda (1986, on p.116).

Title

groupoid C*-convolution algebras

Canonical name

GroupoidCconvolutionAlgebras

Date of creation

2013-03-22 18:13:59

Last modified on

2013-03-22 18:13:59

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

63

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55Q70

Classification

msc 55Q55

Classification

msc 55Q52

Classification

msc 81R50

Classification

msc 22A22

Classification

msc 81R10

Classification

msc 20L05

Classification

msc 18B40

Synonym

convolution algebra

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Defines

groupoid C*-convolution algebra

Defines

groupoid C*-algebra

Defines

groupoid

C

*

-algebra

Defines

Haar systems

Defines

C*-convolution

Defines

measured groupoid

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